Monotonic Decompositions of Submodular Set Functions

Huber and Kolmogorov [Towards minimizing $k$-submodular functions, in Proceedings of ISCO 2012, Lecture Notes in Comput. Sci. 7422, Springer, Heidelberg, 2012, pp. 451--462] introduced a concept of $k$-submodular function as a generalization of ordinary submodular (set) functions and bisubmodular functions and obtained a min-max theorem for the minimization of $k$-submodular functions. Also Kuivinen [Discrete Optim., 8 (2011), pp. 459--477] considered submodular functions on (product lattices of) diamonds and showed a min-max theorem for the minimization of submodular functions on diamonds. In the present paper we consider a common generalization of $k$-submodular functions and submodular functions on diamonds, which we call a transversal submodular function (a t-submodular function, for short). We show a min-max theorem for the minimization of t-submodular functions in terms of a new norm composed of $\ell_1$ and $\ell_\infty$ norms. This reveals a relationship between the obtained min-max theorem and that for the minimization of ordinary submodular set functions due to Edmonds [Submodular functions, matroids, and certain polyhedra, in Proceedings of the Calgary International Conference on Combinatorial Structures and Their Applications, R. Guy, H. Hanani, N. Sauer, and J. Schönheim, eds., Gordon and Breach, New York, 1970, pp. 69--87].We also show how our min-max theorem for t-submodular functions can be used to prove the min-max theorem for $k$-submodular functions by Huber and Kolmogorov and that for submodular functions on diamonds by Kuivinen. Moreover, we show a counterexample to a characterization, given by Huber and Kolmogorov [Towards minimizing $k$-submodular functions, in Proceedings of ISCO 2012, Lecture Notes in Comput. Sci. 7422, Springer, Heidelberg, 2012, pp. 451--462], of extreme points of the $k$-submodular polyhedron and make it a correct one by fixing a flaw therein.

The connections between convexity and submodularity are explored, for purposes of minimizing and learning submodular set functions. First, we develop a novel method for minimizing a particular class of submodular functions, which can be expressed as a sum of concave functions composed with modular functions. The basic algorithm uses an accelerated first order method applied to a smoothed version of its convex extension. The smoothing algorithm is particularly novel as it allows us to treat general concave potentials without needing to construct a piecewise linear approximation as with graph-based techniques. Second, we derive the general conditions under which it is possible to find a minimizer of a submodular function via a convex problem. This provides a framework for developing submodular minimization algorithms. The framework is then used to develop several algorithms that can be run in a distributed fashion. This is particularly useful for applications where the submodular objective function consists of a sum of many terms, each term dependent on a small part of a large data set. Lastly, we approach the problem of learning set functions from an unorthodox perspective---sparse reconstruction. We demonstrate an explicit connection between the problem of learning set functions from random evaluations and that of sparse signals. Based on the observation that the Fourier transform for set functions satisfies exactly the conditions needed for sparse reconstruction algorithms to work, we examine some different function classes under which uniform reconstruction is possible.

Maximization of submodular set functions arises in wireless applications such as scheduling, caching, and leader selection. For a centralized entity with oracle access to the submodular function, submodular maximization can be approximated up to a constant factor using polynomial-time algorithms; such an entity, however, may be unavailable in decentralized wireless networks. In this paper, we consider maximization of a time-varying submodular function by distributed, resource-constrained nodes. We present algorithms for unconstrained distributed submodular maximization, as well as monotone submodular maximization subject to cardinality constraints. For the unconstrained submodular maximization problem, our algorithm achieves an expected optimality gap of 1/3. For cardinality-constrained submodular maximization, our algorithm achieves an expected optimality gap of 1/2, while reducing the storage and communication overhead, as well as the computation requirements of the nodes, compared to existing techniques. We evaluate our approach through an experimental study using sensor scheduling data, and find that our approach is within ten percent of the best achievable utility in the unconstrained case and within five percent in the constrained case.

Abstract: Given a graph G = (V,E) and a function f:2^V -> R (provided by an oracle), the problem [PCliqW] consists in finding a partition into cliques of V(G) of minimum cost. Here, the cost of a partition is the sum of the costs of the cliques in the partition. We provide a polynomial time dynamic program for the case where G is an interval graph and f belongs to a subclass of submodular set functions, which we call This provides a common solution for various generalizations of the coloring problem in co-interval graphs such as max-coloring, Greene-Kleitman's dual, probabilist coloring and chromatic entropy. In the last two cases, this is the first polytime algorithm for co-interval graphs. In contrast, NP-hardness of related problems is discussed. We also describe an ILP formulation for [PCliqW] which gives a common polyhedral framework to express min-max relations such as \overline{\chi}=\alpha for perfect graphs and the polymatroid intersection theorem. This approach allows to provide a min-max formula for [PCliqW] if G is the line-graph of a bipartite graph and f is submodular. However, this approach fails to provide a min-max relation for [PCliqW] if G is an interval graphs and f is value-polymatroidal. Key words: Partition into cliques, Interval graphs, Circular arc graphs, Max-coloring, Probabilist coloring, Chromatic entropy, Partial q-coloring, Batch-scheduling, Submodular functions, Bipartite matchings, Split graphs

We prove a new class of inequalities for submodular set functions, indexed by chordal graphs. Since entropy is a particularly useful example of a submodular function, we deduce some entropy inequalities. As a further corollary, we construct a novel family of determinant inequalities for sums of positive definite Hermitian matrices, and also recover an inequality of Barrett, Johnson, and Lundquist (1989).

Polybasic polyhedra: structure of polyhedra with edge vectors of support size at most 2

We address the problem of maximizing an unknown submodular function that can only be accessed via noisy evaluations. Our work is motivated by the task of summarizing content, e.g., image collections, by leveraging users' feedback in form of clicks or ratings. For summarization tasks with the goal of maximizing coverage and diversity, submodular set functions are a natural choice. When the underlying submodular function is unknown, users' feedback can provide noisy evaluations of the function that we seek to maximize. We provide a generic algorithm — ExpGreedy — for maximizing an unknown submodular function under cardinality constraints. This algorithm makes use of a novel exploration module— TopX — that proposes good elements based on adaptively sampling noisy function evaluations. TopX is able to accommodate different kinds of observation models such as value queries and pairwise comparisons. We provide PAC-style guarantees on the quality and sampling cost of the solution obtained by ExpGreedy. We demonstrate the effectiveness of our approach in an interactive, crowdsourced image collection summarization application.

Joint utility-maximization problems for multi-agent systems often should be addressed by distributed strategy-selection formulation. Constrained by discrete feasible strategy sets, these problems are broadly formulated as NP-hard combinatorial optimization problems. In many cases, these problems can be cast as constrained submodular set function maximization problems, which also belong to the NP-hard domain of problems. A prominent example is the problem of multi-agent mobile sensor dispatching over a discrete domain. This paper considers a class of submodular optimization problems that consist of maximization of a monotone and submodular set function subject to a partition matroid constraint over a group of networked agents that communicate over a connected undirected graph. We work with the value oracle model. Consequently, the only access of the agents to the utility function is through a black box that returns the utility function value given a specific strategy set. We propose a distributed suboptimal polynomial-time algorithm that enables each agent to obtain its respective strategy via local interactions with its neighboring agents. Our solution is a fully distributed gradient-based algorithm using the submodular set functions’ multilinear extension followed by a distributed stochastic Pipage rounding procedure. This algorithm results in a strategy set that when the team utility function is evaluated at the worst case, the utility function value is in 1c(1−e−c−O(1/T)) of the optimal solution with c being the curvature of the submodular function. An example demonstrates our results.

A decomposition theory for submodular functions is described. Any such function is shown to have a unique decomposition consisting of indecomposable functions and certain highly decomposable functions, and the latter are completely characterized. Applications include decompositions of hypergraphs based on edge and vertex connectivity, the decomposition of matroids based on three-connectivity, the Gomory—Hu decomposition of flow networks, and Fujishige’s decomposition of symmetric submodular functions. Efficient decomposition algorithms are also discussed.

Submodular Functions are a special class of Set Functions, which generalize several Information Theoretic quantities such as Entropy and Mutual Information [1]. Submodular functions have subgradients and subdifferentials [2] and admit polynomial time algorithms for minimization, both of which are fundamental characteristics of convex functions. Submodular functions also show signs similar to concavity. Submodular function maximization, though NP hard, admits constant factor approximation guarantees and concave functions composed with modular functions are submodular. In this paper, we try to provide a more complete picture on the relationship between submodularity with concavity. We characterize the superdifferentials and polyhedra associated with upper bounds and provide optimality conditions for submodular maximization using the superdifferentials.

Let Open image in new window be a distributive lattice formed by subsets of a finite set with set union and intersection as the lattice operations, and let f be a submodular function on Open image in new window . The pair ( Open image in new window , f) is called a submodular system and is a generalization of a (poly-)matroid. The present paper makes a survey of the author’s earlier work on submodular systems and provides a unifying view and some useful observations on related topics such as geometries on posets, generalized polymatroids, boundary hypermatroids, submodular functions on crossing families, submodular flows, strongly connected orientations of graphs, Lovasz’s extension of set functions, minimization of submodular functions etc. We also show a new approach to the problem of minimizing submodular functions.

Using polarity, we give an outer polyhedral approximation for the epigraph of set functions. For a submodular function, we prove that the corresponding polar relaxation is exact; hence, it is equivalent to the Lovász extension. The polar approach provides an alternative proof for the convex hull description of the epigraph of a submodular function. Computational experiments show that the inequalities from outer approximations can be effective as cutting planes for solving submodular as well as non-submodular set function minimization problems.

In this paper, for the univariate Bernstein–Kantorovich, Szasz–Mirakjan–Kantorovich and Baskakov–Kantorovich operators written in terms of the Choquet integral with respect to a monotone and submodular set function, we obtain quantitative approximation estimates, uniform and pointwise in terms of the modulus of continuity. In addition, we show that for large classes of functions, the Kantorovich–Choquet type operators approximate better than their classical correspondents. Also, we construct new Szasz–Mirakjan–Kantorovich–Choquet and Baskakov–Kantorovich–Choquet operators, which approximate uniformly f in each compact subinterval of \([0, +\infty )\) with the order \(\omega _{1}(f; \sqrt{\lambda _{n}})\), where \(\lambda _{n}\searrow 0\) arbitrary fast.

Uniform and pointwise convergence of Bernstein–Durrmeyer operators with respect to monotone and submodular set functions

By using the concept of Choquet nonlinear integral with respect to a submodular set function, we introduce the nonlinear Picard–Choquet operators, Gauss–Weierstrass–Choquet operators and Poisson–Cauchy–Choquet operators with respect to a family of submodular set functions. Quantitative approximation results of the order $$\omega _{1}(f; t)_{\mathbb {R}}$$, $$t>0$$, are obtained with respect to the Choquet measure $$\mu (A)=\sqrt{M(A)}$$ where M represents the Lebesgue measure, and with respect to some families of possibility measures. Also, due to the possibilities of choice for the submodular set functions, for some subclasses of functions we prove that these Choquet type operators have essentially better approximation properties than their classical correspondents.